Lie algebra representation

A representation of a Lie algebra $\lag$ is a Lie algebra homomorphism $$\rho:\lag \rightarrow \End V,$$ where $\End V$ is the commutator Lie algebra of some vector space $V$ . In other words, $\rho$ is a linear mapping that satisfies $$\rho([a,b]) = \rho(a)\rho(b)-\rho(b)\rho(a),\quad a,b\in\lag$$ Alternatively, one calls $V$ a $\lag$ -module, and calls $\rho(a),\, a\in \lag$ the action of $a$ on $V$ .

We call the representation faithful if $\rho$ is injective.

A invariant subspace or sub-module $W\subset V$ is a subspace of $V$ satisfying $\rho(a)(W)\subset W$ for all $a\in\lag$ . A representation is called irreducible or simple if its only invariant subspaces are $\{0\}$ and the whole representation.

The dimension of $V$ is called the dimension of the representation. If $V$ is infinite-dimensional, then one speaks of an infinite-dimensional representation.

Given a pair of representations, we can define a new representation, called the direct sum of the two given representations:

If $\rho:\lag\to\End(V)$ and $\sigma:\lag\to\End(W)$ are representations, then $V\oplus W$ has the obvious Lie algebra action, by the embedding $\End(V)\times\End(W)\hookrightarrow\End(V\oplus W)$ .